In this work, we develop an SDPI-based framework for studying hypothesis testing and estimation problems under \((\varepsilon ,\delta )\) -LDP, extending the results of [1]} to approximate LDP. In particular, we derive bounds for both the minimax and Bayesian estimation risks that hold for any \(\varepsilon \ge 0\)  and \(\delta \ge 0\) . Interestingly, when setting \(\delta = 0\) , our bounds can be slightly stronger than [2]}.
